1. Field of the Disclosure
The disclosure is related generally to the field of interpretation of measurements made by well logging instruments for the purpose of determining the properties of earth formations. More specifically, the disclosure is related to a method of interpreting multi-component resistivity measurements made in a bi-axially anisotropic medium.
2. Background of the Art
Electromagnetic induction and wave propagation logging tools are commonly used for determination of electrical properties of formations surrounding a borehole. These logging tools give measurements of apparent resistivity (or conductivity) of the formation that when properly interpreted are diagnostic of the petrophysical properties of the formation and the fluids therein.
It is well known that certain earth formations consist of thin layers of electrically conductive materials interleaved with thin layers of substantially non-conductive material. The response of the typical electromagnetic induction resistivity well logging instrument will be largely dependent on the conductivity of the conductive layers when the layers are substantially parallel to the flow path of the eddy currents. The substantially non-conductive layers will contribute only a small amount to the overall response of the instrument and therefore their presence will typically be masked by the presence of the conductive layers. The non-conductive layers, however, are the ones which are typically hydrocarbon-bearing and are of the most interest to the instrument user. Some earth formations which might be of commercial interest therefore may be overlooked by interpreting a well log made using the electromagnetic induction resistivity well logging instruments known in the art. Such formations are characterized by transverse isotropy and two resistivities, a horizontal resistivity Rh in a plane parallel to the bedding and a vertical resistivity Rv in a direction perpendicular to the bedding.
U.S. Pat. No. 5,999,883 issued to Gupta et al, (the “Gupta patent”), the contents of which are fully incorporated here by reference, discloses a method for determination of an initial estimate of the horizontal and vertical conductivity of anisotropic earth formations. Electromagnetic induction signals induced by induction transmitters oriented along three mutually orthogonal axes are measured. One of the mutually orthogonal axes is substantially parallel to a logging instrument axis. The electromagnetic induction signals are measured using first receivers each having a magnetic moment parallel to one of the orthogonal axes and using second receivers each having a magnetic moment perpendicular to a one of the orthogonal axes which is also perpendicular to the instrument axis. A relative angle of rotation of the perpendicular one of the orthogonal axes is calculated from the receiver signals measured perpendicular to the instrument axis. An intermediate measurement tensor is calculated by rotating magnitudes of the receiver signals through a negative of the angle of rotation. A relative angle of inclination of one of the orthogonal axes which is parallel to the axis of the instrument is calculated, from the rotated magnitudes, with respect to a direction of the vertical conductivity. The rotated magnitudes are rotated through a negative of the angle of inclination. Horizontal conductivity is calculated from the magnitudes of the receiver signals after the second step of rotation. An anisotropy parameter is calculated from the receiver signal magnitudes after the second step of rotation. Vertical conductivity is calculated from the horizontal conductivity and the anisotropy parameter.
U.S. Pat. No. 6,643,589 to Zhang et al., having the same assignee as the present application and the contents of which are incorporated herein by reference, teaches a method for the simultaneous inversion of measurements made by a multi-component logging tool to obtain anisotropic resistivities and formation inclination angle and azimuth. A model that includes horizontal and vertical resistivities is used to generate a simulated tool response. An iterative solution that gives an improved match between the model output and the field observations is obtained using a global objective function. The global objective function is defined as a sum of a data objective function (difference between the model output and the observed data) and a model objective function that stabilizes the inversion procedure by placing a penalty on large changes in the model at each iteration.
U.S. Pat. No. 6,574,562 to Tabarovsky et al. teaches a method of determination of horizontal and vertical conductivities of subsurface formations using a combination of data acquired with a multi-component induction logging tool and data acquired with a conventional high definition induction logging tool. The multi-component data are acquired at a plurality of frequencies and a skin-effect correction is applied. An isotropic resistivity model is derived from HDIL data (multiple frequency or multiple spacing). This may be done either by inversion or by focusing. Using a forward modeling program, expected values of the transverse components of the multi-component data are derived. A skin-effect correction is applied to the model output. Differences between the focused model output and the focused acquired data are indicative of anisotropy and this difference is used to derive an anisotropy factor. Computationally, the method in Tabarovsky is faster than that in Zhang.
The methods discussed in Tabarovsky and in Zhang are effective in analyzing transversely isotropic (TI) media. In a TI medium, resistivity along a symmetry axis is different from resistivity in any direction in a plane orthogonal to the symmetry axis. Certain types of hydrocarbon reservoirs include cross-bedding wherein within geologic markers (or beds) there is, in addition, fine bedding (cross-bedding) with a different dip than the main bedding. Cross bedding typically occur in three major environments: (1) aeolian, (2) subaqueous delta, and (3) river channels. Due to variations in grain size, cementation, water saturation and other factors, cross-bedding usually displays resistivity anisotropy. On a fine scale, the resistivity perpendicular to the cross-bedding planes is in general higher than that parallel to the cross-bedding plane. Accurate reservoir evaluation and description should consider the resistivity dependence with direction. U.S. Pat. No. 7,317,991 to Wang, having the same assignee as the present application and the contents of which are incorporated herein by reference, teaches a method of evaluating such a cross-bedding formation via an approximate, bi-axially anisotropic model. A weak anisotropy approximation is used by Wang. It would be desirable to have a method of evaluating bi-axially anisotropic earth formations without making a weak anisotropy approximation. The present disclosure addresses this need.